Planck's constant | physics | Britannica.com gamma ray Planck’s constant, (symbol h), fundamental physical constant characteristic of the mathematical formulations of quantum mechanics , which describes the behaviour of particles and waves on the atomic scale, including the particle aspect of light . The German physicist Max Planck introduced the constant in 1900 in his accurate formulation of the distribution of the radiation emitted by a blackbody , or perfect absorber of radiant energy (see Planck’s radiation law ). The significance of Planck’s constant in this context is that radiation , such as light , is emitted, transmitted, and absorbed in discrete energy packets, or quanta , determined by the frequency of the radiation and the value of Planck’s constant. The energy E of each quantum , or each photon , equals Planck’s constant h times the radiation frequency symbolized by the Greek letter nu, ν, or simply E = hν. A modified form of Planck’s constant called h-bar (ℏ), or the reduced Planck’s constant, in which ℏ equals h divided by 2π, is the quantization of angular momentum . For example, the angular momentum of an electron bound to an atomic nucleus is quantized and can only be a multiple of h-bar.

Overview of Planck’s constant, including its discovery by Max Planck. Contunico © ZDF Enterprises GmbH, Mainz The dimension of Planck’s constant is the product of energy multiplied by time, a quantity called action . Planck’s constant is often defined, therefore, as the elementary quantum of action. Its value in metre-kilogram-second units is 6.62607004 × 10−34 joule∙second, with a standard uncertainty of 0.000000081 × 10−34 joule∙second. Learn More in these related articles: Can GPS find variations in Planck's constant? - physicsworld.com physicsworld.com Photonic, plasmonic and multiphysics simulations using CST STUDIO SUITE Sponsored by CST of America Watch the video about new electromagnet-based VSMs Lake Shore Cryotronics Learn about the 8600 Series VSMs, which combine high sensitivity, rapid measurement speed and simple operation in a system capable of characterizing a broad range of materials with unprecedented ease – particularly when performing FORC measurements or measuring low-moment materials.

Looking for variations in Planck's constant Physicists in the US say that they have used publicly available data from global positioning system (GPS) satellites to put a limit on how much Planck's constant might vary from place to place. Their technique involves analysing the tiny corrections that are applied to the atomic clocks used in GPS satellites – corrections that are made to account for relativistic effects caused by the orbits of the satellites. However, not all physicists agree that the analysis is meaningful. GPS relies on extremely accurate measurements of how long it takes signals to travel from several satellites to a receiver on the ground. As these timings have to be incredibly precise, atomic clocks are used on board the satellites. These clocks measure the frequency of electromagnetic radiation involved in a specific atomic transition, which is related to its energy via Planck's constant. For reasons outlined in Einstein's special and general theories of relativity, the clocks in space must be corrected on a regular basis so that they agree with atomic clocks on Earth and with each other. Once the appropriate relativistic corrections are made, however, the time on the clocks should be identical regardless of any differences in the local gravitational potentials of the clocks.

So if there is still a deviation once relativity is taken into account, Planck's constant might vary from place to place – in other words, the concept known as local Planck invariance (LPI) might not hold true. James Kentosh and Makan Mohageg of California State University at Northridge now say that they have used these publicly available satellite data to put an upper limit on the possible violation of LPI in the value of Planck's constant. By comparing the corrections that were actually made to the clocks with those defined by relativity, Kentosh and Mohageg have calculated an upper limit on the extent that LPI violation is occurring. Comparing corrections Planck's constant defines the size of energy quanta in a physical system: a quantum of electromagnetic radiation (the photon) has energy proportional to Planck's constant multiplied by the frequency of the electromagnetic field, for example. Kentosh and Mohageg looked at seven of the 32 GPS satellites on the basis that these had the most stable clocks. The clock corrections are published at 15 min intervals and the pair looked at 13 months of data in 2010–2011.

They looked at how the corrections varied as a function of the radial distance of the satellite from Earth and found a small discrepancy in what would be expected from relativity. This, however, is not enough to put a limit on LPI violation for Planck's constant because it is possible that the way time dilation affects the different clocks could also violate LPI. The team says that it has taken this into consideration by looking at a study done in 2010 by Victor Flambaum of the University of New South Wales, and colleagues in Australia and France. Flambaum and the team compared the time kept by an atomic clock with the time on a cryogenic sapphire oscillator clock at the Paris Observatory. According to Kentosh and Mohageg, the results of Flambaum's experiment can be used to place a limit on the LPI violation of time dilation in their study. Putting all of this together, Kentosh and Mohageg find that Planck's constant satisfies LPI to within 0.007. Three faults? However, not everyone agrees with this conclusion, including Flambaum himself. He told physicsworld.com that he finds three faults with a paper describing the analysis, which has been published in Physical Review Letters .

Planck's constant has the dimensions of energy multiplied by time and Flambaum says there is no point in looking for variations in fundamental constants that have dimensions because the units defining those dimensions could also be varying. Instead, he believes that physicists should focus on dimensionless quantities, such as the fine-structure constant. He also points out that the atomic clocks used in the study were all similar and therefore their operations have the same dependencies on physical constants – making it impossible to look for violations of LPI. Instead, he argues that two different clocks are required. Finally, Flambaum does not agree with Kentosh and Mohageg's interpretation of his 2010 clock-comparison paper. Kentosh told physicsworld.com that he and Mohageg are aware of the limitations of their analysis: "It is important to note that our results apply only within the context of general relativity, which predicts that the speed of light, rest mass, energy and length are invariant when measured locally. If any of those macroscopic variables were to vary with gravitational potential, as measured locally, it would skew our findings." Kentosh and Mohageg are now working on a new paper that describes their technique in more detail.

Meanwhile, Flambaum and colleague Julian Berengut have submitted a comment on the topic for publication in Physical Review Letters. Part of the IT standards and organizations glossary: Planck's constant, symbolized h, relates the energy in one quantum ( photon ) of electromagnetic radiation to the frequency of that radiation. In the International System of units ( SI ), the constant is equal to approximately 6.626176 x 10-34 joule-seconds. In the centimeter-gram-second ( cgs ) or small-unit metric system, it is equal to approximately 6.626176 x 10-27 erg-seconds. The energy E contained in a photon, which represents the smallest possible 'packet' of energy in an electromagnetic wave, is directly proportional to the frequency f according to the following equation: E = hf If E is given in joule s and f is given in hertz (the unit measure of frequency), then: E = (6.626176 x 10-34) f and conversely: Also see: electromagnetic field , energy , frequency , SI , and Table of Physical Units and Constants.

See a Khan Academy video, Introduction to light: This was last updated in April 2015 What is Planck's constant? - Quora Quora Written Apr 4, 2016 In quantum physics , the energy of electromagnetic radiation is restricted to indivisible packets called photons . Every photon has an energy [math]hf [/math], [math]f[/math] being the frequency of the radiation , and [math]h[/math] a universal constant recognized by Max Planck in 1900 and called the Planck constant . Here are some more explanations related to this constant : Planck’s constant, (symbol h), fundamental physical constant characteristic of the mathematical formulations of quantum mechanics , which describes the behaviour of particles and waves on the atomic scale, including the particle aspect of light . The German physicist Max Planck introduced the constant in 1900 in his accurate formulation of the distribution of the radiation emitted by a blackbody , or perfect absorber of radiant energy . The significance of Planck’s constant in this context is that radiation, such as light , is emitted, transmitted, and absorbed in discrete energy packets, or quanta, determined by the frequency of the radiation and the value of Planck’s constant.

The energy E of each quantum , or each photon , equals Planck’s constant h times the radiation frequency symbolized by the Greek letter nu, ν, or simply E = hν. A modified form of Planck’s constant called h-bar (ℏ), or the reduced Planck’s constant, in which ℏ equals h divided by 2π, is the quantization of angular momentum . For example, the angular momentum of an electron bound to an atomic nucleus is quantized and can only be a multiple of h-bar. The dimension of Planck’s constant is the product of energy multiplied by time, a quantity called action . Planck’s constant is often defined, therefore, as the elementary quantum of action. Its value in metre-kilogram-second units is [math]6.62607004 × 10^{-34}[/math] joule∙second, with a standard uncertainty of [math]0.000000081 ×10^{-34}[/math] joule∙second. Written Sep 3 The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of action , central in quantum mechanics .

First recognized in 1900 by Max Planck , it was originally the proportionality constant between the minimal increment of energy , E, of a hypothetical electrically charged oscillator in a cavity that contained black body radiation, and the frequency , f, of its associated electromagnetic wave . In 1905, the value E, the minimal energy increment of a hypothetical oscillator, was theoretically associated by Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called the photon . The Planck–Einstein relation connects the particulate photon energy E with its associated wave frequency f In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz ) it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant or Dirac constant.

It is equal to the Planck constant divided by 2π, and is denoted ħ (pronounced "h-bar") 88 Views · Answer requested by Updated Jan 31, 2016 To explain further, from what Himanshu has rightly said : The energy of a particle is given by the Einstein's theory, where the particle has a mass of 'm' in grams as its unit However when we are dealing with a photon or light particle, the element and unit of mass is not understood. Hence Max Planck gave a different solution. As per his theory, the energy of a wave increases with the velocity of the wave but is inversely proportional to its wave length. But the velocity of an electromagnetic radiation is limited to the speed of light (c), which is a constant at 30,000,000,000 cm / sec. Hence the wave length is what matters or the frequency of the wave "v". More the wave length lesser the energy and vice versa. The proportionality is removed by adding a proportionality constant 'h' in the equation as mentioned below.

The equation of Max Planck giving the relationship between the Energy, speed of light and the wavelength of light was a break through in Quantum Physics and hence the 'Proportionality Constant 'h' has been named after him. Further it is to be noted that even the mass of a photon, can be deducted from the above equations of Einstein and Max Planck. In the above equations, The symbols used are in Red and the Units of the symbols in Blue. Finally note how the length of units (cm) in the final equation gets cancelled to arrive at the unit for Planck's constant. Planck constant Facts for Kids | KidzSearch.com Illustration taken from Newton's original letter to the Royal Society (1 January 1671 [Julian calendar]). S represents sunlight. The light between the planes BC and DE are in colour. These colours are recombined to form sunlight on the pane GH Young's double slit experiment Between 1670 and 1900 scientists discussed the nature of light. Some scientists believed that light consisted of many millions of tiny particles. Other scientists believed that light was a wave . [2] Light - waves or particles?

In 1678 Christiaan Huygens wrote the book Traité de la lumiere. He believed that light was made up of waves. He said that light could not be made up of particles because light from two beams do not bounce off each other. [3] [4] In 1672 Isaac Newton wrote the book Opticks. He believed that light was made up of red, yellow and blue particles which he called corpusles. Newton explained this by his "two prism experiment". The first prism broke light up into different colours. The second prism merged these colours back into white light. [5] [6] [7] During the Eighteenth Century Newton's theory was given the most attention. [8] In 1803 Thomas Young described the "double-slit experiment" . [4] In this experiment light from two narrow slits interfere with each other. This caused a pattern which showed that light was made up of waves. For the rest of the nineteenth century wave theory of light was given the most attention. In the 1860's James Clerk Maxwell developed equations that described electromagnetic radiation as waves.

The theory of electromagnetic radiation treats light, radio waves, microwaves and many other types of wave as the same thing except that they have different wavelengths. The wavelength of light is between 400 and 600 nm, [Note 1] the wavelength of radio waves varies from 10 m to 1500 m and the wavelength of microwaves is about 2 cm. In a vacuum , all electromagnetic waves travel at the speed of light . The frequency of the electromagnetic wave is given by: ν The symbols are defined here . Black body radiators All hot bodies give off radiant heat . Radiant heat is electromagnetic radiation. Normally this radiation is in the infra-red range, but if the body is very hot (1000 °C or more), it is in the visible range. In the late 1800's many scientists studied wavelength of electromagnetic radiation from black-body radiators at different temperatures. Rayleigh-Jeans Law Rayleigh-Jeans curve and Planck's curve plotted against photon wavelength. Lord Rayleigh first published the basics of the Rayleigh-Jeans law in 1900. The theory was based on the Kinetic theory of gasses .

Sir James Jeans published a more complete theory in 1905. The law relates the quantity and wavelength of electromagnetic energy given off by a black body radiator at different temperatures. The equation describing this is: [9] B − 1 {\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}} . The symbols are defined here . If the wavelength of light is allowed to become very large, then it can be shown that the Raleigh-Jeans and the Planck relationships are almost identical. He found that his equations only worked if the energy of the light quanta were related to frequency by the relationship: where He calculated h and k and found that h = 6.55×10−27 erg·sec. k = 1.34×10−16 erg·deg-1. The values are close to the modern day accepted values of 6.62606×10−27 and 1.38065×10−16 respectively.

The Planck law agrees well with the experimental data, but its full significance was only appreciated several years later. Quantum theory of light Solway Conference 1911. Planck , Einstein and Jeans are standing. Planck is second from the left. Einstein is second from the right. Jeans is fifth from the right. Wien is seated, third from the right It turns out that electrons are dislodged by the photoelectric effect if light reaches a threshold frequency. Below this no electrons can be emitted from the metal. In 1905 Albert Einstein published a paper explaining the effect. Einstein proposed that a beam of light is not a wave propagating through space, but rather a collection of discrete wave packets ( photons ), each with energy. Einstein said that the effect was due to a photon striking an electron. This demonstrated the particle nature of light. Einstein also found that there electromagnetic radiation with a long wavelength had no effect. Einstein said that this was because the "particles" did not have enough energy to disturb the electrons. [11] [12] [13] Einstein realised that the energy of each photon was related to the photon frequency by the Planck constant.

This could be written mathematically as: E {\displaystyle E=h\nu ={\frac {hc}{\lambda }}} . The symbols are defined here . In 1921 Einstein received the Nobel Prize for linking the Planck constant to the photoelectric effect. Application The Planck constant is of importance in many applications. A few are listed below. Bohr model of the atom Bohr's model of the atom. An electron falling from the n=3 shell to the n=2 shell loses energy. This energy is carried away as a single photon. Visible spectrum of Neon. Each line represents a different pair of energy levels. In 1913 Niels Bohr published the Bohr model of the structure of an atom. Bohr said that the angular momentum of the electrons going around the nucleus can only have certain values. These values are given by the equation L {\displaystyle L=n{\frac {h}{2\pi }}} where n = positive integer . h = Planck constant. The Bohr model of the atom can be used to calculate the energy of electrons at each level. Electrons will normally fill up the lowest numbered states of an atom.

If the atom receives energy from, for example, an electric current, electrons will be excited into a higher state. The electron will then drop back to a lower state and will loose their extra energy by giving off a photon. Because the energy levels have specific values, the photons will have specific energy levels. Light emitted in this way can be split into different colours using a prism. Each element has its own pattern. The pattern for neon is shown alongside. Heisenberg's uncertainty principle In 1927 Werner Heisenberg published the uncertainty principle . The principle states that it is not possible to make a measurement without disturbing the thing being measured. It also puts a limit on the minimum disturbance caused by making a measurement. In the macroscopic world these disturbances make very little difference. For example, if the temperature of a flask of liquid is measured, the thermometer will absorb a small amount of energy as it heats up. This will cause a small error in the final reading, but this error is small and not important. In quantum mechanics things are different. Some measurements are made by looking at the pattern of scattered photons . One such example is Compton scattering .

If both the position and momentum of a particle is being measured, the uncertainty principle states that there is a trade-off between the accuracy with which the momentum is measured and the accuracy with which the position is measured. The equation that describes this trade-off is: Δ h = Planck constant. Colour of light emitting diodes Simple LED circuit that illustrates use of the Planck constant. The colour of the light emitted is depends on the voltage drop across the diode. The wavelength of the light can be calculated using the Planck constant. [14] In the electric circuit shown on the right, the voltage drop across the light emitting diode (LED) depends on the material of the LED. For silicon diodes the drop is 0.6 V. However for LEDs it is between 1.8 V and 2.7 V. This information enables a user to calculate the Planck constant. [14] The energy needed for one electron to jump the potential barrier in the LED material is given by E 2.62 Value of the Planck constant and the kilogram redefinition Since its discovery, measurements of h have become much better.

Planck first quoted the value of h to be 6.55×10−27 erg·sec. This value is within 5% of the current value. As of 3 March 2014, the best measurements of h in SI units is 6.62606957×10−34 J·s. The equivalent figure in cgs units is 6.62606957×10−27 erg·sec. The is relative uncertainty of h is 4.4×10−8. The reduced Planck constant (ħ) is a value that is sometimes used in quantum mechanics . It is defined by ℏ {\displaystyle \hbar ={\frac {h}{2\pi }}} . Planck units are sometimes used in quantum mechanics instead of SI. In this system the reduced Planck constant has a value of 1, so the value of the Planck constant is 2π. Plancks constant can now be measured with very high precision. This has caused the BIPM to consider a new definition for the kilogram. They have proposed that the international prototype kilogram is no longer used to define the kilogram.

Instead the BIPM will define the Planck constant to have an exact value. Scientist will use this value and the definitions of the metre and the second to define the kilogram. [15] Notes photons - Why do we use Planck's constant? - Physics Stack Exchange Why do we use Planck's constant? up vote 9 down vote favorite 6 I have been trying to reason why energy packets (i.e. photons ) are assumed to be quantized. I know this originated from Max Planck, but may someone explain why energy couldn't be emitted continuously rather than in packets? Also: Planck said he didn't understand the reason for this, but it happens. So, how did he arrive at the assumption? Thanks for any clarification! up vote 7 down vote The fundamental reason is that light waves at temperature T have a definite length scale, a typical wavelength, and classical electromagnetic theory has a scaling invariance which forbids such a scale from emerging. In the classical theory, the energy in electromagnetic waves just leaks into ever smaller distances. The leaking can be understood by thermal arguments.

If you have a certain amount of energy in a gas, you can distribute this energy in many ways--- you can put all the energy in one molecule, and have the other molecules sitting still, or you can distribute it roughly equally. The roughly equal distribution is vastly more likely. The average kinetic energy of a gas molecule is one definition of the thermodynamic temperature $T$, and in thermal equilibrium, all molecules have roughly the same kinetic energy, with a probability distribution determined completely just from the energy of each motion. The same is true for fields. If you have a certain amount of energy in an electromagnetic field, you can distribute it over long-wavelength oscillations only, or over all oscillations equally. By probability considerations, you would prefer the maximally broad distribution. Since there are infinitely many modes at short distances, the classical energy will partition itself into little tiny increments, and leak to short distance electromagnetic jiggles, and this is called the ultraviolet catastrophe: all energy in the universe goes into the shortest wavelengths of light, and everything else stops and freezes. This is not what is observed in nature. Rather, we see that at any temperature T there is a preferred wavlength $\lambda$ of emitted light.

So that as a stove gets hotter, it gets red, then yellow, then white, then blue, then ultraviolet. It doesn't leak energy into short-wavelengths until T is at least a certain amount. This means that there is something preventing energy from going into short wavelengths. What is it? Whatever it is, it must include a physical constant with dimensions mixing length and energy, which can convert from temperature (average kinetic energy) to wavelength. Since all light in thermal equilibrium with a gas looks the same, no matter what the gas or what the materials of the wall is, it must be a universal constant of nature. Similarly, you have all sorts of parts inside an atom, like electrons, or nuclei. If these things could jiggle around, the energy would distribute into these too, and all atoms would be jiggling around with a lot of extra energy. This is not observed--- atoms and small molecules don't jiggle. This is the paradox of specific heats noted by Maxwell in the 1870s. These puzzles prevented full acceptance of Boltzmann's idea of statistical entropy.

Wien's adiabatic law The breakthrough came in 1898, when Wien noted that the distribution of thermal light in a container has to be consistent with the principle of adiabatic invariance. This is the principle that the thermal equilibrium state, if you change the parameters slowly, stays a thermal equilibrium state with the same entropy. From this, Wien was led to study what happens to light in a box of mirrors when you move one of the mirrors inward slowly. A moving mirror doesn't reflect light like a stationary mirror, it does some work against the radiation pressure. Further, light reflected from a moving mirror is slightly bluer than the light that comes in. So the energy and the frequency of the reflected light are both changed. If you want to check this today, you can see it is true by considering photon reflection. Wien showed this for classical EM waves, and the justification of photons comes from this reasoning, not the other way around. Although the energy and the frequency are both changed when you move the mirror inward, their ratio stays the same.

This ratio is an adiabatic invariant: $$ J= {E\over f}$$ This relation showed Wein that whatever the correct thermal equilibrium law, it must have this property: if you take the probability distribution of energy at any one frequency, shift the frequency by a certain amount, then the energy should change by the same amount. This means that, up to a scaling and normalization, the probability distribution is a function of the adiabatic invariant $E/f$ Wein then guessed that this probability distribution is exponential in E, since Boltzmann had shown that at temperature T, the probability of a classical state with energy E goes like $e^{-\beta E}$. This led Wien to guess the form $e^{-b{E\over f}}$ for the probability of having energy E at frequency f, where the constant $b$ is a new constant of nature. In units where Boltzmann's constant $k$ is set to 1, so that energy and temperature are measured with the same units, b is just the reciprocal of Planck's constant. Wien's law explains the emergence of a length scale.

The typical energy determines a typical frequency $f = bkT$, and determines a typical wavelength by $\lambda= {c\over f}$. Wien's form is not correct except at high frequencies. At low frequencies, you find that the energy in each mode is not exponentially distributed, but on average equal to $kT$, like for any harmonic oscillation. Planck interpolated the two forms with a clever guess, and tried to justify the interpolation. His reasoning is not optimal, since he was motivated more by experimental data, and he wasn't sure of Boltzmann's hypothesis, nor did he appreciate the ultraviolet catastrophe fully (at least not according to Kuhn, who examined the original papers in detail--- I don't know if Kuhn is right, I just skimmed the original paper). I will follow Einstein's 1905 reasoning, which starts where Wien left off. Einstein calculated the entropy of Wien's distribution, assuming it is correct, and noticed that the entropy at each frequency goes like $$ Nlog({V\over N})$$ Where $N= {bE\over f}$ will be seen to be the number of photons at frequency f.

The form of the entropy is counting the number of possible configurations of high-frequency light. The number of configurations is the exponential of the entropy, and it goes like $$ {V^N\over N!} $$ it's the entropy of an ideal gas. This suggests strongly that the light is composed of photons, localizable particles each carrying an energy $hf$ where $h$ is Planck's constant, the reciprocal of Wein's constant $b$. The adiabatic invariance of the thermal photon gas means that the photons have to be able to bounce off slowly moving walls and stay single photons, and consistency with classical EM means that as the frequency of a photon changes, the energy changes proportionally, so that the photon must carry an energy proportional to the frequency, and the proportionality constant is a universal constant of nature. From this hypothesis, Einstein went and solved the specific heat problem. He also was able to resolve the paradoxes of light emission in relativity, and deduce $E=mc^2$.

It was Einstein who created the modern photon concept, and pretty much he alone believed in photons until 1919, when Milliken established experimentally that the photoelectric effect goes as Einstein says, and Compton showed that photons can individually collide with electrons in the famous Compton scattering experiments. Once you know that there are bundles of energy of size $hf$, the ideas of Boltzmann allow you to rederive Planck's distribution law for thermal light, by assuming that the probability of having N photons at frequency f goes as $e^{-\beta Nhf}$ where $\beta=1/kT$ is the inverse temperature, and $Nhf$ is the total energy the $N$ photons have together. While the deductive path is clearest through Einstein, it was Planck who came up with the distribution first. Planck's reasoning is based on interpolation, and I wouldn't be able to reproduce it logically, because it involves an unjustified assumption that the specific heat of the photon gas (or something like that, I don't remember what) would be the average of the Wien law and the Stephan-Boltzmann law. Required math: algebra, calculus, complex numbers Required physics: kinetic energy Reference: Griffiths, David J.

(2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Chapter 1. Although it is perfectly acceptable to begin a study of quantum mechanics by simply stating the Schrödinger equation, it is interesting to have a look at some of the motivation for it. Before we do, however, it is important to realize that the Schrödinger equation cannot be derived from any other underlying physical principles. It is a statement of how physics works and can be viewed as a postulate that must be accepted without proof. In that sense, it plays the same role as the equation in Newton’s view of physics, and the same role as Maxwell’s four equations in the classical theory of electromagnetism. All of these physical theories were written down in an attempt to explain experimental results; none of them was derived from any deeper physical theory (although it is possible, of course, that it may be possible to derive them from some deeper theory that we have not yet discovered, but so far that hasn’t happened).

Although there are a fair number of experiments whose explanation requires some modification to classical physics, the ideas that gave rise to Schrödinger’s equation came from the observation that the behaviour of light (or more precisely, electromagnetic radiation, which includes light and other radiation such as X-rays and radio waves) in some cases requires that we assume it to be composed of particles (photons). Max Planck’s original proposal was that the energy of a photon was directly proportional to its frequency, which he wrote as where (the Greek letter ‘nu’) is the frequency of the photon in cycles per second, and is the constant of proportionality, now known as Planck’s constant for obvious reasons. A wave of any sort is characterized by three quantities: its frequency , or number of times per second a peak of the wave passes a fixed point, its wavelength , or distance between two adjacent peaks, and its velocity , or the speed at which one particular peak moves (try not to get and confused). Note that these three quantities are related: That is, the speed of a wave is its frequency multiplied by its wavelength. A wave with a particular wavelength can move faster, so that more peaks pass a fixed point in a given time; thus its frequency is larger.

One of the consequences of the theory of special relativity, devised by Einstein and first published in 1905, is the famous equation , but what is not often realized is that the in the equation is the particle’s apparent mass, which depends on its speed relative to the observer. If a particle is at rest relative to the observer, then where is the rest mass of the particle. If the particle is moving relative to the observer, then it will appear to have a momentum and one of the predictions of relativity is the energy of a moving particle is given by the equation: Note that if the particle is at rest relative to the observer, then it has zero momentum ( ), and the equation reduces to at it should. Now notice that if the particle has zero rest mass ( ), then the energy becomes According to relativity, a massless particle is the only type of particle that can travel at the speed of light and in fact a massless particle must travel at the speed of light at all times. (By the way, it is sometimes said that light travels more slowly when passing through a dense transparent substance such as glass. Although it may take a beam of light more time to pass through the glass, the individual photons still travel at speed inside the glass.

They have to bounce between the atoms of the glass in order to pass through it, so it is this process which appears to slow the light beam down.) So we now have two very simple equations from which we can calculate the energy of a photon. One gives the energy in terms of the frequency; the other gives the energy in terms of the velocity. However, since the velocity is related to the frequency as we saw above, we can rewrite the first equation as Comparing this with the second equation, we get So far, all these calculations have been done for photons (which is why the velocities involved have all been equal to c). In 1924, Louis de Broglie proposed (in his PhD thesis no less) that all particles can be represented by waves and that the formula for the momentum of a photon also applies to all particles, whether or not they have rest mass. That is, he proposed that for all particles. At this point we should pause for a moment and modify these equations slightly to bring them into line with what is usually seen in textbooks. Rather than use the frequency (which is expressed in cycles per second) it is more usual to use the frequency measure (which is expressed in radians per second).

With this alteration, the energy-frequency equation becomes where The momentum relation is usually written using a quantity called the wave number, so the momentum-wavelength equation becomes Now, in non-relativistic physics, the energy of a free particle (that is, a particle not being acted on by any force) is entirely kinetic , and can be written as since the momentum of a particle is . Equating this to the other expression for the energy, we get This is a relation between the frequency and the wave number (and thus with the wavelength ) and in classical wave theory is known as a dispersion relation, since it describes how rapidly a localized packet of waves with various wavelengths (or wave numbers) will disperse over time. Remember that the velocity is and for particles that have rest mass, the velocity can be anything less than the speed of light, so the relation between the frequency and wave number need not be constant. Thus waves with different frequencies can travel at different velocities and with a dispersion relation like we have here, they will travel at different velocities. You could be forgiven for thinking that we haven’t got much closer to writing down the Schrödinger equation, but in fact there isn’t much more to do.

Since we want to represent particles as waves, we need to think up an equation whose solution is a wave of some sort. (Remember, we’re jumping from postulate to postulate at this stage; there is no rigorous mathematical derivation.) The simplest form of wave is that represented by a sine or cosine wave, so if we want a wave with a particular frequency and wave number, we can postulate a function (in one dimension) of form where is the amplitude (height) of the wave. If you’re not convinced this represents a wave, you can do a few tests on it. The cosine repeats after every change of in its argument, so if we fix the time at, say, and then measure the distance between successive peaks, we get . Now remember that the wave number is related to the wavelength by , so the distance between two peaks is found from so the distance between two peaks is indeed one wavelength. For the frequency, we can fix our sights on one position, say , and then measure how long ( ) it takes for one complete wave to pass that point. The first complete wave will have passed when . Since so the number of complete wavelengths to pass the point in one second is which again is correct.

Finally, we can measure the velocity of the wave by looking at one fixed point on the wave. That is, we can examine a point on the wave where for some constant . If we take the derivative with respect to of this equation, we get so the velocity of the wave is . Now in classical physics, it is common to use complex numbers to represent a wave by using the relation with the understanding that the part that represents the wave is always the real part of the equation. If we try the same trick here, we can represent a wave by the quantity Now, finally, we are ready to get to the Schrödinger equation. What we want, then is a differential equation with a wave-like solution given by above that also has the dispersion relation . There are various equations that could satisfy these requirements, but the simplest one appears to be the Schrödinger equation for a free particle: We can check that above satisfies this equation by taking its derivatives: Plugging these back into the Schrödinger equation we get so the dispersion relation is satisfied. That’s really all there is to it.

If it looks like a series of kludges, that’s because it is; the physicists who contributed the various stages in the argument really were groping in the dark and postulating various equations and conditions and hoping for the best. In fact, there was one more kludge which Schrödinger made to get the final form of the equation that is satisfied by a particle moving under the influence of a potential, and that was just to add the potential term onto the left hand side, so the final form is One final note about this equation. While the use of complex numbers in classical physics was just a computational convenience with the understanding that the function that described the physical wave was just the real part, in quantum mechanics, the wave function must be a complex function because of the that multiplies the right hand side of the Schrödinger equation. This leads to the consequence that itself cannot represent a physically measurable quantity, a consequence that was resolved by Max Born’s assumption that the square modulus of the function (which is real) represented the probability density for finding the particle at a particular location and time.

Introduction to the Fundamental Physical Constants home page Introduction to the constants for nonexperts The following material is intended to give nonexperts insight into the general subject of fundamental physical constants. It is adopted from an article written by Barry N. Taylor in about 1971 for the 15th edition of the Encyclopaedia Britannica © 1974 , and is reproduced with permission. Although a considerable portion of the article is now out-of-date, the introductory and historical portions given here remain generally valid. Note, however, that the terms accuracy and uncertainty are used in the text in a manner that is not always consistent with current practice. This also applies to other terms such as parts per million and its symbol ppm, and the now obsolete terms atomic weight and molecular weight. To add a more modern flavor to the material, a fundamental constant experiment of current interest is also described: the determination of the fine-structure constant by means of the quantum Hall effect and calculable cross capacitor. Introduction ( index ) Throughout all of the formulations of the basic theories of physics and their application to the real world, there appear again and again certain fundamental invariant quantities.

These quantities, called the fundamental physical constants, and which have specific and universally used symbols, are of such importance that they must be known to as high an accuracy as is possible. They include the velocity of light in vacuum (c); the charge of the electron, the absolute value of which is the fundamental unit of electric charge (e); the mass of the electron (me); Planck's constant (h); and the fine-structure constant, symbolized by the Greek letter alpha. These will all be considered in detail below. There are, of course, many other important quantities that can be measured with high accuracy -- the density of a particular piece of silver, for example, or the lattice spacing (the distance between the planes of atoms) of a particular crystal of silicon, or the distance from the Earth to the Sun. These quantities, however, are generally not considered to be fundamental constants. First, they are not universal invariants because they are too specific, too closely associated with the particular properties of the material or system upon which the measurements are carried out.

Second, such quantities lack universality because they do not consistently appear in the basic theoretical equations of physics upon which the entire science rests, nor are they properties of the fundamental particles of physics of which all matter is constituted. It is important to know the numerical values of the fundamental constants with high accuracy for at least two reasons. First, the quantitative predictions of the basic theories of physics depend on the numerical values of the constants that appear in the theories. An accurate knowledge of their values is therefore essential if man hopes to achieve an accurate quantitative description of the physical universe. Second, and more important, the careful study of the numerical values of these constants, as determined from various experiments in the different fields of physics, can in turn test the overall consistency and correctness of the basic theories of physics themselves. Definition, importance, and accuracy The constants named above, five among many, were listed because they exemplify the different origins of fundamental constants. The velocity of light (c) and Planck's constant (h) are examples of quantities that occur naturally in the mathematical formulation of certain fundamental physical theories, the former in James Clerk Maxwell's theory of electric and magnetic fields and Albert Einstein's theories of relativity, and the latter in the theory of atomic particles, or quantum theory.

For example, in Einstein's theories of relativity, mass and energy are equivalent, the energy (E) being directly proportional to the mass (m), with the constant of proportionality being the velocity of light squared (c2) -- i.e., the famous equation E = mc2. In this equation, E and m are variables and c is invariant, a constant of the equation. In quantum theory, the energy (E) and frequency, symbolized by the Greek letter nu ( ), of a photon (a single quantum unit of electromagnetic energy such as light or heat radiation) are related by E = h . Here, Planck's constant (h) is the constant of proportionality. The elementary charge (e) and the electron mass are examples of constants that characterize the basic, or elementary, particles that constitute matter, such as the electron, alpha particle, proton, neutron, muon, and pion. Additionally, they are examples of constants that are used as standard units of measurement.

The charge and mass of atomic and elementary particles may be expressed in terms of the elementary charge (e) and the electron mass (me); the charge of an alpha particle, the nucleus of the helium atom, is given as 2e, whereas the mass of the muon is given as 206.77 me. The fine-structure constant (alpha) is an example of a fundamental constant that can be expressed as a combination of other constants. The fine-structure constant is equal to a numerical constant times the velocity of light times the elementary charge squared divided by twice Planck's constant, or µ0ce2/2h, µ0 being the so-called permeability of free space, numerically equal to exactly 4 x 10-7. (The system of measurement units used in this article is the Système International d'Unités [ International System of Units ], or SI.) Because this particular combination of constants always appears in theoretical equations in exactly the same way, however, the fine-structure constant is really a fundamental constant in its own right.

For example, the fine-structure constant is the fundamental constant of quantum electrodynamics, the quantum theory of the interaction (mutual influence) among electrons, muons, and photons. As such it is a measure of the strength of these interactions. Another quantity that is a combination of other constants is the Rydberg constant (symbolized Rinfinity), which is equal to the product µ02c3e4me/8h3. It sets the scale (magnitude) of the various allowed electron energy states or levels in atoms such as hydrogen. The accuracy with which many of the fundamental constants can be currently measured is a few parts in a million. By accuracy is meant the relative size of the uncertainty that must be assigned to the numerical value of any quantity to indicate how far from the true value it may be because of limitations in experiment or theory. This uncertainty is a quantitative estimate of the extent of the doubts associated with the value. The most commonly used uncertainty, the standard deviation, symbolized by the Greek letter sigma, is such that there is about a 68 percent chance that the true value lies within plus or minus sigma.

Furthermore, there is a 95 percent chance that the true value lies between plus and minus two standard deviations and a 99.7 percent chance that it lies between plus and minus 3 standard deviations. (All uncertainties quoted in this article will be one standard deviation.) In practice, an uncertainty of one part per million (abbreviated ppm) is rather respectable. It corresponds to determining the length of a United States football field (100 yards, or about 91 meters) to within the thickness of two of these pages (one page is about 0.0022 inch or 0.056 millimeter thick). There are several quantities that have been measured with uncertainties approaching one part in 1 000 000 000 000 (one in 1012); this uncertainty corresponds to determining the distance from New York to San Francisco to within one-tenth the thickness of a piece of paper. quantum mechanics - What is the point of the reduced Planck's constant $\hbar$ (h-bar)? - Why don't we just have Planck's Constant $h$?

- Physics Stack Exchange Do we have a question about "natural units" motivated by the ground state properties of the hydrogen atom? Quite a few useful microscopic quantities come close to 1 when they are being measured as multiples of the hydrogen scale rather than in m, s, J etc.. – CuriousOne Dec 17 '14 at 18:03 1 This is actually very similar to the use of circumflex ^ and umlaut ¨ which originally came about because parchment was expensive and they were shorthands for frequent letter combinations - e.g. ae -> ä, us -> û. $\hbar$ is just another shorthand. – Floris Dec 17 '14 at 19:59 @KyleKanos Yes. Normally we are just happy with setting $\hbar = c = 1$. The nice thing about that is, once you have a result, you can go back to regular units by adding enough $c$ and $\hbar$s to get the units right :-). – safkan Dec 17 '14 at 20:12 @KyleKanos: Exactly. Just wondering if we had a question about that, already?

– CuriousOne Dec 18 '14 at 1:46 @KyleKanos: Thanks! I was wondering what the search syntax for this kind of search was. Now I don't have to ask anymore and can do it myself. I appreciate the help! – CuriousOne Dec 18 '14 at 2:00

Question: In physics, which letter is used to represent Planck's Constant?
Answer: H